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How ψ-epistemic models fail at explaining the
indistinguishability of quantum states
Cyril Branciard
To cite this version:
Cyril Branciard. How ψ-epistemic models fail at explaining the indistinguishability of quantum
states. Physical Review Letters, American Physical Society, 2014, Physical Review Letters, 2 (113),
pp.020409. �10.1103/PhysRevLett.113.020409�. �hal-01091208�
Cyril Branciard
School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia Institut N´eel, CNRS and Universit´e Grenoble Alpes, 38042 Grenoble Cedex 9, France
(Dated: July 11, 2014)
We study the extent to which ψ-epistemic models for quantum measurement statistics—models where the quantum state does not have a real, ontic status—can explain the indistinguishability of nonorthogonal quantum states. This is done by comparing the overlap of any two quantum states with the overlap of the corresponding classical probability distributions over ontic states in a ψ-epistemic model. It is shown that in Hilbert spaces of dimension d ≥ 4, the ratio between the classical and quantum overlaps in any ψ-epistemic model must be arbitrarily small for certain nonorthogonal states, suggesting that such models are arbitrarily bad at explaining the indistinguishability of quantum states. For dimensions d = 3 and 4, we construct explicit states and measurements that can be used experimentally to put stringent bounds on the ratio of classical-to-quantum overlaps in ψ-epistemic models, allowing one in particular to rule out maximally ψ-epistemic models more efficiently than previously proposed.
Despite its central role, the quantum state remains one of the most mysterious objects of quantum theory. Does it correspond to any physical reality, or does it merely represent one’s information on a quantum system? These questions have triggered intense debates among physi-cists and philosophers since the advent of quantum the-ory, and are still the subject of active research in the study of quantum foundations.
The general framework of ontological models [1] pro-poses a rigorous approach to address such questions. This framework presupposes the existence of underlying states of physical reality—ontic states—and describes the quan-tum state as a state of knowledge—an epistemic state— about the actual ontic state of a given quantum system, represented by a probability distribution over the set of ontic states. A fundamental distinction is made between so-called ψ-ontic models, in which any underlying on-tic state determines the quantum state uniquely, and so-called ψ-epistemic models, where the same ontic state can be compatible with different quantum states; i.e., the probability distributions corresponding to two dif-ferent quantum states can overlap. In the former case, the ontic state “encodes” the quantum state, which can hence be understood as a physical property of the sys-tem, while in the latter case the quantum state cannot be given the status of a real physical property.
The epistemic view of the quantum state is quite at-tractive, as it gives a natural explanation to many puz-zling quantum phenomena [2]—including, for instance, the collapse of the wave function, or the impossibility to perfectly distinguish nonorthogonal quantum states. However, it has been shown that ψ-epistemic models must be severely constrained if they are to reproduce the statistics of quantum measurements. Notably, Pusey, Barrett, and Rudolph showed that no ψ-epistemic models satisfying some natural independence condition for com-posite systems can reproduce quantum predictions [3]. A
number of other no-go theorems have since been proven, under various assumptions [4–18]. Explicit ψ-epistemic models have nevertheless been constructed for quantum measurement statistics in Hilbert spaces of any dimen-sion [14, 19], which get around the assumptions of these no-go theorems.
While it is thus impossible to completely rule out ψ-epistemic models for quantum theory without auxiliary assumptions, one can still set bounds on the extent to which ψ-epistemic models can possibly explain certain quantum features, and, in particular, the indistinguisha-bility of nonorthogonal quantum states [9–11, 16, 18]. This can be done by investigating how much of the over-lap of two quantum states can be accounted for by the classical overlap of their corresponding epistemic states. In this respect, Maroney and Leifer [9–11] showed that a certain (asymmetric) measure of overlap of classical probability distributions had to be smaller than |hφ|ψi|2
for some quantum states |ψi, |φi in any Hilbert space of dimension d ≥ 3. Barrett et al. [16] showed that the ratio between two directly comparable measures for the clas-sical and quantum overlaps had to scale at most like 1/d for certain pairs of quantum states when the dimension d increases, while Leifer [18] exhibited states for which the same ratio has to decrease exponentially with d. Fol-lowing these works, and in particular the approach of Barrett et al. [16], we show in this Letter that for all di-mensions d ≥ 4, there actually exist states for which the ratio of classical-to-quantum overlaps must be ily small—meaning that ψ-epistemic models are arbitrar-ily bad at explaining the indistinguishability of certain nonorthogonal quantum states in dimensions d ≥ 4. For dimensions 3 and 4, we exhibit explicit states and mea-surements that would allow one to bound experimentally the ratio of classical-to-quantum overlaps, in a more effi-cient way than previously proposed [16].
2 Ontological models.— Let us start by recalling the
framework of ontological models [1, 20]. Such models posit the existence of ontic states λ, taking values in some measurable space Λ, meant to describe the real state of affairs of a given physical system [21]. The preparation of a quantum state |ψi corresponds to the preparation of an ontic state λ according to some probability distribution— called epistemic state—µψ(λ), such that
µψ(λ) ≥ 0 and
Z
Λ
µψ(λ) dλ = 1. (1)
In the following, the notation ψ will be used to refer to both the quantum state |ψi and the corresponding epistemic state µψ.
When a measurement M is performed, the probability for each of its possible outcomes m is supposed to de-pend only on the ontic state, and is given by a response function ξM(m|λ). For the response function to be a
well-defined probability distribution for each ontic state λ, it must be nonnegative and normalized; i.e., it satisfies
ξM(m|λ) ≥ 0 and
X
m
ξM(m|λ) = 1. (2)
When the measurement is performed on a system pre-pared in the epistemic state µψ, the probability of each
outcome is given, after integrating over Λ, by PM(m|ψ) =
Z
Λ
ξM(m|λ) µψ(λ) dλ. (3)
The model reproduces the quantum measurement statis-tics in a d-dimensional Hilbert space H if for any state |ψi ∈ H and any projection eigenbasis M = {|mi} of H, the above probabilities PM(m|ψ) are those predicted
by quantum theory—namely, according to the Born rule, PM(m|ψ) = |hm|ψi|2.
In this framework, an ontological model is said to be ψ-epistemic if there exists at least one pair of different (pure) quantum states |ψi, |φi, for which the correspond-ing probability distributions µψ, µφ have nonzero
over-lap; otherwise, the model is said to be ψ-ontic. Hence, in a ψ-epistemic models a single ontic state λ could be compatible with more than one epistemic state. This suggests that the impossibility of perfectly distinguish-ing two nonorthogonal quantum states may be (at least partially) explained by the fact that the underlying real state of affairs may sometimes be the same for the two states.
To quantify this, we shall compare, as in Refs. [12, 16, 18], the probability of successfully distinguishing the two quantum states |ψi, |φi using optimal quantum mea-surements, and that of distinguishing the two epistemic states µψ, µφ given that one knows the ontic state λ
(as-suming in each case that |ψi and |φi, respectively µψ
and µφ, have been prepared with equal probabilities).
These probabilities of success are given, respectively, by
1 − ωQ(|ψi, |φi)/2 and 1 − ωC(µψ, µφ)/2, where the
quan-tum and classical overlaps ωQ and ωC are defined as [16]
ωQ(|ψi, |φi) = 1 − p 1 − |hψ|φi|2, (4) ωC(µψ, µφ) = Z Λ minµψ(λ), µφ(λ) dλ. (5)
Clearly, one has 0 ≤ ωC(µψ, µφ) ≤ ωQ(|ψi, |φi) ≤ 1
in any model that reproduces quantum measurement statistics [12]. Indeed, given λ one can reproduce the optimal quantum measurement that gives a probability 1 − ωQ(|ψi, |φi)/2 of distinguishing the two preparations;
the optimal classical strategy for distinguishing µψ and
µφ cannot give a lower probability of success.
Ontolog-ical models such that ωC(µψ, µφ) = ωQ(|ψi, |φi) for all
states ψ, φ are said to be maximally ψ-epistemic [22]: these models fully explain the indistinguishability of nonorthogonal quantum states by that of the correspond-ing epistemic states. In general, for any two nonorthog-onal states ψ, φ (such that ωQ(|ψi, |φi) 6= 0) we shall
define the ratio of classical-to-quantum overlaps as κ(ψ, φ) = ωC(µψ, µφ)
ωQ(|ψi, |φi)
≤ 1. (6)
As the quantum and classical overlaps have the same operational interpretation in terms of probabilities of successful distinctions, this ratio suitably quantifies how much the model explains of the quantum indistinguisha-bility of |ψi and |φi.
Constraints on ψ-epistemic models.— We now show how the possible values of κ(ψ, φ) are constrained for ontological models that reproduce quantum predictions. Let us start by introducing a first Lemma; its proof fol-lows closely that of Ref. [16], and is given in Appendix A: Lemma 1. Consider n + 1 states {ψj}0≤j≤n (with n ≥
2). For each triplet (ψ0, ψj1, ψj2), with 1 ≤ j1< j2≤ n,
consider a measurement Mj1,j2 with 3 possible outcomes
(m0, m1, m2). Then (with j0= 0) X 1≤j≤n ωC(µψ0, µψj) = X 1≤j≤n κ(ψ0, ψj) ωQ(|ψ0i, |ψji) ≤ 1 + X 1≤j1<j2≤n 2 X i=0 PMj1,j2(mi|ψji). (7)
An interesting situation is when for each triplet of states (ψ0, ψj1, ψj2), there exists a measurement
Mj1,j2 such that PMj1,j2(m0|ψ0) = PMj1,j2(m1|ψj1) =
PMj1,j2(m2|ψj2) = 0—in which case the quantum states
(|ψ0i, |ψj1i, |ψj2i) are said to be PP-incompatible [23].
In such a situation the left-hand side of Inequality (7) is simply 1. If the states |ψji are furthermore such
that all quantum overlaps ωQ(|ψ0i, |ψji) are equal (i.e.,
ωQ(|ψ0i, |ψji) = ωQ(|ψ0i, |ψ1i) for all 1 ≤ j ≤ n), then
Lemma 1 implies 1 n X 1≤j≤n κ(ψ0, ψj) ≤ 1 n ωQ(|ψ0i, |ψ1i) , (8)
which in turn implies that at least one term κ(ψ0, ψj) in the average above is upper-bounded by
1/[n ωQ(|ψ0i, |ψ1i)]. Refs. [16] and [18] exhibited states
for which the above upper bound on the ratios of classical-to-quantum overlaps must decrease like 1/d and exponentially, respectively, with the Hilbert space dimen-sion d (see Appendix B). We significantly improve these results by showing here the existence of states, in any dimension d ≥ 4, with arbitrarily low values of κ. For that we shall make use of the following Lemma:
Lemma 2. In any Hilbert space H of dimension d ≥ 3, and for any n ≥ 2, there exist n + 1 quantum states {|ψj(n)i}0≤j≤nsuch that
(i) for all 1 ≤ j ≤ n, ωQ(|ψ (n) 0 i, |ψ (n) j i) = 1 − q 1 − 14n−1/(d−2) > 1 8n −1/(d−2);
(ii) all triplets of states (|ψ0(n)i, |ψ(n)j
1 i, |ψ
(n)
j2 i), for 1 ≤
j1< j2≤ n, are PP-incompatible.
Proof. Let us fix an arbitrary state |ψ0(n)i ∈ H. From the study of optimal Grassmannian line packings in the (d−1)-dimensional subspace orthogonal to |ψ0(n)i, one can show that it is possible to find n states {|φ(n)j i}1≤j≤n
in this orthogonal subspace such that for all j1 6= j2,
|hφ(n)j
1 |φ
(n) j2 i|
2 ≤ 1 − n−1/(d−2) [24]. From these n states
|φ(n)j i we define, for 1 ≤ j ≤ n and with χ = 1 4n
−1/(d−2)
(such that 0 < χ < 14), the states |ψj(n)i =√χ |ψ0(n)i + √
1−χ |φ(n)j i, which indeed satisfy (i). For 1 ≤ j1 < j2 ≤ n, defining x1 = |hψ (n) 0 |ψ (n) j1 i| 2, x2 = |hψ (n) 0 |ψ (n) j2 i| 2 and x 3 = |hψ (n) j1 |ψ (n) j2 i| 2, one has x1 = x2 = χ and x3 = χ + (1−χ)hφ(n)j 1 |φ (n) j2 i 2 ≤ χ + (1−χ)|hφ(n)j 1 |φ (n) j2 i| 2 ≤ χ + (1−χ)√1 − n−1/(d−2)2 ≤ 1−2χ2
. One then finds that x1 + x2 + x3 ≤ 2χ +
(1−2χ)2 < 1 and (1−x
1−x2−x3)2 ≥ 4χ2(1−2χ)2 ≥
4x1x2x3, which shows that (|ψ (n) 0 i, |ψ (n) j1 i, |ψ (n) j2 i) are PP-incompatible [16, 23].
The states just constructed thus satisfy the assump-tions of Eq. (8), which gives
1 n X j κ(ψ0(n), ψj(n)) ≤ 1 n1−q1−14n−1/(d−2) < 8/nd−3d−2. (9) In particular, for each n, there exists at least one state ψ(n)j such that κ(ψ0(n), ψ(n)j ) ≤ 8/nd−3d−2. Noting that for
any (fixed) d ≥ 4, this upper-bound tends to 0 as n → ∞, we conclude that, as claimed before:
Theorem 3. For any ontological model reproducing quantum measurement statistics in a Hilbert space of di-mension d ≥ 4, there exist (nonorthogonal) states φ, ψ
with an arbitrarily low ratio of classical-to-quantum over-laps κ(ψ, φ).
It remains an open question whether the same result holds for a Hilbert space of dimension d = 3 or not. On the other hand, in the case of dimension d = 2, one can check that the Kochen-Specker model [1, 25] for projec-tive measurements is maximally ψ-epistemic—i.e., is such that κ(ψ, φ) = 1 for all nonorthogonal states ψ, φ.
Upper-bounding κ(ψ, φ) experimentally. — The ar-gument leading to Theorem 3 above required us to con-sider infinitely many states (n+1 → ∞) and measure-ments (in n(n−1)2 → ∞ bases). Nevertheless, using finitely many states and measurements already allows one to put nontrivial bounds on some ratios of classical-to-quantum overlaps κ(ψ, φ), which can be verified experi-mentally.
In order to simplify the discussion below, we will now assume (as in Ref. [16]) that in the ontological model un-der study, κ(ψ, φ) is lower-bounded by a fixed value κ0:
κ(ψ, φ) ≥ κ0for all ψ, φ. Our goal will then be to
upper-bound the value of κ0. Note, however, that in all we write
below, κ0can simply be replaced by min1≤j≤n[κ(ψ0, ψj)]
for the states under consideration—or even, when all quantum overlaps ωQ(|ψ0i, |ψji) are equal, by the
av-erage n1P
jκ(ψ0, ψj) (as in Eq. (8)).
In regard to experimental tests, one may want to minimize the number of states and measurements to be used, as well as the bound on κ0. Although the
PP-incompatibility property used above for each triplet (|ψ0(n)i, |ψ(n)j
1 i, |ψ
(n)
j2 i) was quite convenient to obtain our
theoretical results, using states with such a property is in fact not optimal for experimental purposes. Note indeed that this PP-incompatibility criterion is not necessary in our approach, and neither is the assumption that all quantum overlaps ωQ(|ψ0i, |ψji) are equal: it is still
pos-sible without those to bound the value of κ0 by using
Eq. (7) of Lemma 1, which implies that
κ0 ≤ 1 +P j1<j2 P2 i=0 PMj1,j2(mi|ψji) P jωQ(|ψ0i, |ψji) . (10)
The right-hand side above can be obtained experimen-tally by preparing the n + 1 quantum states |ψji and
measuring them according to some measurements Mj1,j2,
so as to estimate the probabilities PMj1,j2(mi|ψji). Note
that no assumption needs to be made on the measure-ments to calculate the bound (they can in principle cor-respond to any unknown POVM); one, however, needs to make sure that the states |ψji are reliably prepared,
so that the quantum overlaps ωQ(|ψ0i, |ψji) are reliably
calculated.
Already for a Hilbert space of dimension d = 3, it is possible to derive nontrivial bounds for κ0. The authors
of Ref. [16] constructed a set of n+1 = 9+1 states {|ψji}
4 that using n+1 = 3+1 states is actually sufficient in
prin-ciple to get a nontrivial bound, although the bound we obtained is quite close to 1, and may be hard to demon-strate experimentally: for the states and measurements we found, one gets, from (10), κ0 ≤ 0.9964 (see
Ap-pendix C). Moving to n + 1 = 4 + 1, we obtained, with the states and measurements specified in Appendix C, the bound κ0 ≤ 0.9361—which already improves (using
fewer states and measurements, i.e. fewer experimental resources) the bound given in Ref. [16], thus allowing one to rule out a strictly larger class of ψ-epistemic models for quantum measurement statistics.
An important consideration for experimental tests is the resistance to noise. In order to get a rough estimate of how robust our results are, let us assume that the prob-abilities PMj1,j2(mi|ψji) in Eq. (10) are each estimated
up to a quantity (taken for simplicity to be the same for all these probabilities). In order to take the worst case into account in the bound of Eq. (10), we thus add to each term PMj1,j2(mi|ψji)—that is, we add 3 ×
n(n−1) 2
in total to the numerator of the fraction in (10). The bound remains nontrival if it is lower than 1; for the states and measurements specified in Appendix C, this requires < 5 × 10−3, which is also more robust than what was found in Ref. [16].
Increasing the number of states (and measurement bases) allows one to decrease the bound on κ0.
Nu-merically we could obtain a bound κ0 ≤ 0.6408 using
n + 1 = 20 + 1 states [26]. We expect it is possible to set an even lower bound by using more states; how low the bound can be, and in particular whether it can be taken down to zero (for n → ∞, as in the case d ≥ 4), is an open question. Note also that as n becomes large, the robustness to noise in general decreases [27].
In dimension d = 4, we found again that n + 1 = 3 + 1 is enough to set a nontrivial bound on κ0 (we actually
found the same bound as in the d = 3 case above). With n + 1 = 4 + 1 states, we could find an improvement over the d = 3 case, obtaining the bound κ0 ≤ 0.9054
(Ap-pendix C). Regarding the robustness to noise, the states and measurements we use require a value of < 7 × 10−3 to give a nontrivial bound on κ0. We recall that, from
the result of Theorem 3, using more states allows one to set an arbitrarily low upper bound on κ0. Namely, any
bound κ0 ≤ ¯κ, for any ¯κ > 0, can be obtained from (9)
by using n ≥ (8/¯κ)d−2d−3 states (for d ≥ 4) constructed
as in Lemma 2 (which may, however, not be optimal). Again, as the number of states gets larger, the resistance to noise decreases—e.g., the states used in (9) require a noise parameter . 1/ 12 nd−1d−2 for the bound to be
lower than 1.
Discussion. — We have proven in this Letter that for any ontological model reproducing the quantum mea-surement statistics in a Hilbert space of dimension d ≥ 4, there exist states ψ, φ whose ratio of classical-to-quantum
overlaps is arbitrarily small. Recalling that this ratio quantifies how much the model explains of the indistin-guishability of the quantum states |ψi and |φi, we con-clude that ψ-epistemic models must be arbitrarily bad in general at explaining quantum indistinguishability.
Note that the states |ψ(n)j i (1 ≤ j ≤ n) used in our proof become closer and closer, as n increases, to be-ing orthogonal to |ψ(n)0 i. Recalling that |hψ0(n)|ψj(n)i|2 = 1
4n
−1/(d−2), what is actually shown by Eq. (9) is the
ex-istence of states ψ, φ such that
κ(ψ, φ) ≤ 4
d
8 |hψ|φi|
2(d−3) with |hψ|φi| → 0. (11)
It has been argued, for reasons to do with coarse grain-ing of measurements, that it may be natural to expect κ(ψ, φ) → 0 when |hψ|φi| → 0 [18, 28]. Our result thus imposes some constraint on the possible scaling of κ(ψ, φ) as the two states tend to orthogonal states, for any ψ-epistemic model reproducing quantum measure-ment statistics—e.g., those of Refs [14, 19]. Another question worth investigating would be to see what bounds on κ(ψ, φ) can be derived for some fixed values of |hψ|φi|. Interestingly, our theoretical result here (as well as that of Ref. [3] and many of the other no-go theorems that fol-lowed) does not depend on the specific form of the Born rule. By considering PP-incompatible triplets of states, we just used in our proof the fact that when quantum theory assigns zero probability to a measurement out-come, the ontological model must do the same [29]. It would be interesting to see if, when the Born rule predicts a nonzero probability, its specific form imposes stronger constraints on possible ψ-epistemic models for quantum measurement statistics (e.g., if it allows one to prove our result for d = 3 as well). This may give more insights on which types of models can reproduce quantum theory.
Using finitely many states and measurements, one can only set finite bounds on the ratios of classical-to-quantum overlaps. We exhibited states and measure-ments allowing one to put nontrivial bounds already for d = 3. These can be used experimentally to rule out some families of ψ-epistemic models—namely, those which as-sign larger ratios of classical-to-quantum overlaps than the bound demonstrated, for the states used experimen-tally (recall that κ0 used for simplicity above can be
re-placed by min1≤j≤n[κ(ψ0, ψj)]). In particular, as soon as
the bound is lower than 1, maximally ψ-epistemic models are ruled out. It is worth emphasizing that this can, in principle, be done without introducing any auxiliary as-sumption, like the independence condition used, e.g., in the experimental test reported in Ref. [12]. Furthermore, the allowed amount of noise for the states and measure-ments exhibited in this Letter for dimensions d = 3 and 4, should make an experiment feasible with current tech-nology, e.g., in photonics.
Acknowledgments. — I acknowledge financial sup-port from a Discovery Early Career Researcher Award (DE140100489) of the Australian Research Council and from the ‘Retour Post-Doctorants’ program (ANR-13-PDOC-0026) of the French National Research Agency.
Appendix
A. Proof of Lemma 1
The proof of Lemma 1 is quite similar to that given in Appendix 1 of Ref. [16]. For completeness and clarity, we present it here explicitly.
For any epistemic state µψ(a nonnegative, normalised
function over the ontic state space Λ), let us define the set
Ψ = {(λ, x) ∈ Λ × R+| 0 ≤ x ≤ µψ(λ)}. (A1)
Denoting by dν = dλ × dx the volume measure on the space Λ × R+
, where dx is the Lebesgue measure of R+,
the volume of the set Ψ is, by normalisation of µψ,
ν(Ψ) = Z Ψ dν = Z Λ Z µψ(λ) 0 dxdλ = Z Λ µψ(λ) dλ = 1. (A2) For two epistemic states µψ0 and µψj, one has
Ψ0∩ Ψj = {(λ, x) | 0 ≤ x ≤ min[µψ0(λ), µψj(λ)]}, (A3) and hence ν(Ψ0∩ Ψj) = Z Λ min[µψ0(λ), µψj(λ)] dλ = ωC(µψ0, µψj). (A4) For now three epistemic states µψ0, µψj1 and µψj2 one
gets, as above, ν(Ψ0∩ Ψj1∩ Ψj2) = Z Λ min[µψ0(λ), µψj1(λ), µψj2(λ)] dλ. (A5) Consider a measurement Mj1,j2 with 3 possible
out-comes (m0, m1, m2). From the normalisation condition of
Eq. (2), we have that for each λ,P2
i=0 ξMj1,j2(mi|λ) = 1.
From (A5), it then follows that (with j0= 0)
ν(Ψ0∩ Ψj1∩ Ψj2) = Z Λ 2 X i=0 ξMj1,j2(mi|λ) min i0 [µψji0(λ)] dλ ≤ 2 X i=0 Z Λ ξMj1,j2(mi|λ) µψji(λ) dλ = 2 X i=0 PMj1,j2(mi|ψji). (A6)
From the n + 1 states {ψj}0≤j≤n, let us define, for
1 ≤ j ≤ n, the n measurable sets
Aj = Ψ0∩ Ψj. (A7)
The second Bonferroni inequality applied to these sets writes ν [ j Aj ≥ X j ν Aj − X j1<j2 ν Aj1∩ Aj2. (A8)
Using Eqs. (A4) and (A6) (noting that Aj1∩ Aj2 = Ψ0∩
Ψj1∩Ψj2) and the fact that ν
S
jAj = ν SjΨ0∩Ψj ≤
ν Ψ0 = 1, this implies that
X j ωC(µψ0, µψj) ≤ 1 + X j1<j2 2 X i=0 PMj1,j2(mi|ψji). (A9)
Note that this inequality holds for any ontological model, independently of whether it reproduces quantum measurement statistics or not. Quantum theory enters when we write ωC(µψ0, µψj) = κ(ψ0, ψj) ωQ(|ψ0i, |ψji)
in the l.h.s. of Inequality (7) of Lemma 1.
B. Comparison with the results of Refs. [16, 18]
Here we compare our results with the bounds obtained in Refs. [16, 18] on the ratios of classical-to-quantum overlaps. For simplicity let us assume that there exists κ0 such that κ(ψ, φ) ≥ κ0 for all ψ, φ, recalling that κ0
can simply be taken to be min1≤j≤n[κ(ψ0, ψj)] for the
states under consideration below—or even, as all quan-tum overlaps ωQ(|ψ0i, |ψji) will here be equal, the
aver-age n1P
jκ(ψ0, ψj).
In Ref. [16], the n = d2 quantum states {|ψ
ji}1≤j≤d2
considered in our analysis were chosen to be the states of d mutually unbiased bases in a Hilbert space of a prime power dimension d ≥ 4, while |ψ0i was taken to
be one of the d states of the remaining mutually unbi-ased basis (recall that for a prime power d, there exist d + 1 mutually unbiased bases [30]). These states sat-isfy ωQ(|ψ0i, |ψji) = 1 −p1 − 1/d for all 1 ≤ j ≤ d2.
Furthermore, for 1 ≤ j1< j2≤ d2, the triplets of states
(|ψ0i, |ψj1i, |ψj2i) thus chosen are all PP-incompatible.
As in Ref. [16], one can then conclude from Eq. (8) of the main text that for d ≥ 4 a prime power,
κ0 ≤
1
d2(1 −p1 − 1/d) <
2
d. (A10) This was furthermore shown to also imply a bound arbi-trary for an dimension d, namely κ0< 4/(d − 1) [16].
In Ref. [18], Leifer followed a different approach to also set bounds on the ratios of classical-to-quantum over-laps. Nevertheless, the states he considered can also be
6 used with our approach. Namely, for a Hilbert space
of any dimension d ≥ 3, let us define the state |ψ0i =
(1, 0, ..., 0)T, and let the n = 2d−1 states {|ψji}1≤j≤2d−1
be the Hadamard states √1
d(1, ±1, ..., ±1)
T. One again
has in this case ωQ(|ψ0i, |ψji) = 1 −p1 − 1/d for all
1 ≤ j ≤ 2d−1. Furthermore, noting that |hψ 0|ψj1i| 2 = |hψ0|ψj2i| 2= 1/d and |hψ j1|ψj2i| 2 ≤ (1 − 2/d)2, one can
check (using the criterion of Refs. [16, 23]) that all triplets of states (|ψ0i, |ψj1i, |ψj2i), for 1 ≤ j1 < j2 ≤ 2
d−1, are
still PP-incompatible. Eq. (8) then leads to a bound on κ0that decreases exponentially with the dimension d:
κ0 ≤
1
2d−1(1 −p1 − 1/d) <
4d
2d. (A11)
This bound actually scales better than that of Ref. [18] (κ0≤ 2d(1−/2)d for dimensions d divisible by 4, where
is a positive constant smaller than 1).
Note that the exponential scaling just obtained re-quires one to consider an exponential number of states. If one also considers an exponential number of states—say, n = 2mfor some m ≥ 1—in Lemma 2 of the main text, one gets from Eq. (9) a bound κ0< 8/2
d−3
d−2m which also
gives and exponential scaling (as m increases), already for d = 4. (For the same number of states as above, i.e. m = d − 1, the states of Lemma 2 give yet a slightly bet-ter scaling with dimension than (A11).) From our study it appears that the scaling of the bound on κ0 as the
number of states increases, or the scaling of κ(ψ, φ) as the inner product |hψ|φi| tends to 0 (see Discussion in the main text), are more relevant than their scaling with the dimension of the Hilbert space.
C. Explicit states and measurements for experimental tests
(for d = 3, 4 and n = 3, 4)
We give here the best states and measurements we found for bounding the value of κ0, in dimension d = 3
using n + 1 = 3 + 1 and n + 1 = 4 + 1 states, and in dimension d = 4 using n + 1 = 4 + 1 states (for the case n + 1 = 3 + 1 in dimension d = 4, the best states and measurements we found are the same as in the qutrit case). The Supplemental Material of Ref. [26] also con-tains the best states and measurements we found for the case d = 3, n+1 = 20+1, which give a bound κ0≤ 0.6408
and a robustness to noise < 10−3. Note that the states and measurements we obtained are the result of a numer-ical search, and are not claimed to be optimal.
The states are written below as ket vectors, while the measurements are given in the form of unitary matri-ces representing a projection eigenbasis: each column of Mj1,j2labelled by i = 0, ..., d−1 corresponds to the
eigen-state |mj1,j2
i i, giving outcome mi. Note that for the case
d = 4, the last column (i = 3) of each matrix Mj1,j2
is always orthogonal to the three states |ψ0i, |ψj1i and
|ψj2i, and hence the corresponding outcome never
ap-pears when the measurement is performed on one of these states. To rigorously define a 3-outcome measurement, we can arbitrarily set for instance m3 = m0, so that
the three measurement operators are {|mj1,j2
0 ihm j1,j2 0 | + |mj1,j2 3 ihm j1,j2 3 |, |m j1,j2 1 ihm j1,j2 1 |, |m j1,j2 2 ihm j1,j2 2 |}. — d = 3, n = 3 —
As claimed in the main text, using n + 1 = 3 + 1 states is sufficient to get nontrivial bounds on κ0.
We shall use here the shorthand notations ck= cos θk
and sk= sin θk. The best states we found are given by
|ψ0i = 1 0 0 , |ψ1i = c1 s1 0 , |ψ2i = c2 s2c3 s2s3 , |ψ3i = c2 s2c3 −s2s3 ,
with θ1= 1.1945, θ2= 0.2839 and θ3= 1.8423. For the
following measurement bases:
M1,2 = c4 s4c6 s4s6 s4c5 −c4c5c6− s5s6 −c4c5s6+ s5c6 −s4s5 c4s5c6− c5s6 c4s5s6+ c5c6 , M1,3 = c4 s4c6 s4s6 s4c5 −c4c5c6− s5s6 −c4c5s6+ s5c6 s4s5 −c4s5c6+ c5s6 −c4s5s6− c5c6 , M2,3 = c7 s7/ √ 2 s7/ √ 2 −s7 c7/ √ 2 c7/ √ 2 0 −1/√2 1 /√2 , with θ4 = 1.6276, θ5 = 2.2192, θ6 = 0.3100 and θ7 =
1.4269, the bound obtained from Eq. (10) of the main text is found to be κ0≤ 0.9964.
When the probabilities appearing in Eq. (10) are esti-mated up to a quantity , as considered in the main text, the upper bound remains non-trivial in the worst case as long as 1+ P j1<j2 P2 i=0PMj1 ,j2(mi|ψji)+32n(n−1) P jωQ(|ψ0i,|ψji) < 1, i.e. when < P jωQ(|ψ0i, |ψji) − 1 −Pj1<j2 P2 i=0PMj1,j2(mi|ψji) 3 2n(n − 1) . (A12) For the above states and measurements, we obtain the requirement that < 6 × 10−4.
— d = 3, n = 4 —
The following n + 1 = 4 + 1 states:
|ψ0i = 1 0 0 , |ψ1i = cθ sθ 0 , |ψ2i = cθ 0 sθ , |ψ3i = cθ −sθ 0 , |ψ4i = cθ 0 −sθ ,
with cθ = cos θ, sθ = sin θ and θ = 0.7152, and the
following measurement bases:
M1,2 = cϕ sϕ/ √ 2 sϕ/ √ 2 sϕ/ √ 2 −(1+cϕ)/2 (1−cϕ)/2 sϕ/ √ 2 (1−cϕ)/2 −(1+cϕ)/2 , M1,3= 0 1/√2 1/√2 0 −1/√2 1/√2 1 0 0 , M1,4 = cϕ sϕ/ √ 2 sϕ/ √ 2 sϕ/ √ 2 −(1+cϕ)/2 (1−cϕ)/2 −sϕ/ √ 2 −(1−cϕ)/2 (1+cϕ)/2 , M2,3 = cϕ sϕ/ √ 2 sϕ/ √ 2 −sϕ/ √ 2 −(1−cϕ)/2 (1+cϕ)/2 sϕ/ √ 2 −(1+cϕ)/2 (1−cϕ)/2 , M2,4= 0 1/√2 1/√2 1 0 0 0 −1/√2 1/√2 , M3,4 = cϕ sϕ/ √ 2 sϕ/ √ 2 −sϕ/ √ 2 (1+cϕ)/2 −(1−cϕ)/2 −sϕ/ √ 2 −(1−cϕ)/2 (1+cϕ)/2 ,
with cϕ = cos ϕ, sϕ = sin ϕ and ϕ = 1.4436, give,
from Eq. (10), the bound κ0 ≤ 0.9361. Note that as
the states |ψji (1 ≤ j ≤ 4) considered here all have
the same quantum overlap with |ψ0i, this bound
actu-ally holds for the average ratio of classical-to-quantum overlaps, 1
4
P
jκ(ψ0, ψj) (see main text).
Regarding the robustness to experimental noise, the requirement on the quantity , calculated as above, is found to be < 5 × 10−3, i.e. can be an order of magnitude larger than in the previous n = 3 case. (Note, besides, that the states and measurements that give the highest robustness to noise may in general be slightly different from those that optimize the bound on κ0 in
the noise-free case.)
— d = 4, n = 4 —
The following n + 1 = 4 + 1 states:
|ψ0i = 1 0 0 0 , |ψ1i = cθ sθ/ √ 3 sθ/ √ 3 sθ/ √ 3 , |ψ2i = cθ sθ/ √ 3 −sθ/ √ 3 −sθ/ √ 3 , |ψ3i = cθ −sθ/ √ 3 sθ/ √ 3 −sθ/ √ 3 , |ψ4i = cθ −sθ/ √ 3 −sθ/ √ 3 sθ/ √ 3 ,
with now θ = 0.7274, and the following measurement bases: M1,2= cϕ sϕ/ √ 2 sϕ/ √ 2 0 sϕ −cϕ/ √ 2 −cϕ/ √ 2 0 0 −1/2 1/2 1/√2 0 −1/2 1/2 −1/√2 , M1,3= cϕ sϕ/ √ 2 sϕ/ √ 2 0 0 −1/2 1/2 1/√2 sϕ −cϕ/ √ 2 −cϕ/ √ 2 0 0 −1/2 1/2 −1/√2 , M1,4= cϕ sϕ/ √ 2 sϕ/ √ 2 0 0 −1/2 1/2 1/√2 0 −1/2 1/2 −1/√2 sϕ −cϕ/ √ 2 −cϕ/ √ 2 0 , M2,3= cϕ sϕ/ √ 2 sϕ/ √ 2 0 0 −1/2 1/2 1/√2 0 1/2 −1/2 1/√2 −sϕ cϕ/ √ 2 cϕ/ √ 2 0 , M2,4= cϕ sϕ/ √ 2 sϕ/ √ 2 0 0 −1/2 1/2 1/√2 −sϕ cϕ/ √ 2 cϕ/ √ 2 0 0 1/2 −1/2 1/√2 , M3,4= cϕ sϕ/ √ 2 sϕ/ √ 2 0 −sϕ cϕ/ √ 2 cϕ/ √ 2 0 0 −1/2 1/2 1/√2 0 1/2 −1/2 1/√2 ,
with now ϕ = 1.4946, give, from Eq. (10), the bound κ0≤
0.9054. Again, as the states |ψji (1 ≤ j ≤ 4) considered
here all have the same quantum overlap with |ψ0i, this
bound actually holds for the average 14P
jκ(ψ0, ψj).
The requirement on the noise parameter is found here to be < 7 × 10−3; for d = 4, an experiment with the above n + 1 = 4 + 1 states would thus allow for some larger noise than one with the previous qutrit states.
8
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